The Magnetic Field from Cylindrical Arc Coils and Magnets: A Compendium with New Analytic Solutions for Radial Magnetization and Azimuthal Current

Abstract

This study provides analytic solutions for the magnetic field of coils and magnets that have a non-axisymmetric cylindrical geometry with a rectangular cross-section. New analytic solutions are provided for radially magnetized permanent magnet arcs, thin coil disc sectors, and thick coil sectors. If components of the 3D field are not representable in closed-form or as canonical Legendre elliptic integrals, the exact solution is given in terms of a series of regularized beta functions. The limit and hence spatial convergence is found to these series, giving a well-defined and fast solving algorithm for computation. The equations can be readily applied to find the magnetostatic field in linear or non-linear systems that contain a large set of elements. Example applications are provided to demonstrate how the field can be used to calculate forces and benchmark computational efficiency of the equations. A thorough review of the preceding literature and background theory is provided before a detailed methodology obtaining the analytic solutions contained in this compendium, and further related geometries in cylindrical or spherical coordinates. This is the first study to comprehensively solve the field equations for this collection of electromagnetic geometries.